Parallelogram Law of Vector Addition
State Parallelogram Law of vector addition
If a parallelogram can be drawn so that two vectors can be placed with their tails connected as the two adjacent sides of the parallelogram with an angle θ between them, then the diagonal of the parallelogram represents their resultant vector or the vector sum.
Derive the formulas of the resultant vector using the Parallelogram Law of vector addition
![Parallelogram Law diagram](https://physicsteacher.in/wp-content/uploads/2021/07/image-41.png)
Let A and B be the two vectors and let θ be the angle between them as shown in Fig. above. To calculate the vector sum, we complete the parallelogram.
Here side PQ represents vector A, side PS represents B and the diagonal PR represents the resultant vector R. Here, α angle is the angle the resultant makes with the base vector and the angle denotes the direction of the resultant or the vector sum.
![Parallelogram Law formula sets - magnitude of the resultant formula & Angle made by the resultant formula](https://physicsteacher.in/wp-content/uploads/2021/07/image-42.png)
Sample Numerical to find the sum of 2 force vectors using the Parallelogram Law
Force A has a magnitude of 30 N and it makes an angle of 30 degrees with another force B of magnitude 40 N. What is the vector sum or resultant of forces A and B?
Solution:
Let, R be the vector sum of forces A and B.
|R| = √[302 + 402 + 2.30.40 cos 30] = 67.66 N
If resultant force R makes an angle α with A then tan α = (B sinθ)/(A + Bcosθ)
here, A = 30, B = 40, θ = 30.tanα = 40 sin 30 / (30 + 40 cos 30) = 20/(30 + 34.64) = .31
α = arctan (.31) = 17.22 degree
Answer: The resultant force is 67.66 N at 17.22 degrees from force A